CUET · MATHS · PYQ PAPER 2023

The point on the curve \(y^2=x\), where the tangent makes an angle of \(\frac{\pi}{4}\) with x-axis is:

A \(\left(\frac{1}{2}, \frac{1}{4}\right)\)
B \(\left(\frac{1}{4}, \frac{1}{2}\right)\)
C (4,2)
D (2,4)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\left(\frac{1}{4}, \frac{1}{2}\right)\)

Step-by-step Solution

Detailed explanation

\(\frac{d}{dx}(y^2) = \frac{d}{dx}(x)\) \(2y \frac{dy}{dx} = 1\) \(\frac{dy}{dx} = \frac{1}{2y}\) \(\text{Slope} = \tan(\frac{\pi}{4}) = 1\) \(\frac{1}{2y} = 1 \Rightarrow y = \frac{1}{2}\) \(x = y^2 = (\frac{1}{2})^2 = \frac{1}{4}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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From CUET

match List - I with List - II

List - I	List - II
(A) Equations of line through (5, -4, 6) with direction ratios 3, 7, 2	(I) \(\frac{x+3}{5}=\frac{y+7}{-4}=\frac{z+2}{6}\)
(B) Equations of line through (3, 7, 2) with direction ratios 5, -4, 6	(II) \(\frac{x-3}{5}=\frac{y-7}{-4}=\frac{z-2}{6}\)
(C) Equations of line through (-5, 4, -6) with direction ratios 3, 7, 2	(III) \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\)
(D) Equations of line through (-3, -7, -2) with direction ratios 5, -4, 6	(IV) \(\frac{x+5}{3}=\frac{y-4}{7}=\frac{z+6}{2}\)

Choose the correct answert from the option given below :

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